Abstract

This paper connects the practice of wireless communication with the mathematics of quadratic forms developed by Radon and Hurwitz about a hundred years ago. Orthogonal designs, known as space-time block codes in the communications literature, provide the bridge between the two subjects. The columns of the design represent different time slots, the rows represent different transmit antennas, and the entries are the symbols to be transmitted. Multiple transmit antennas provide independent paths from the base station to the mobile terminal, and in effect this creates a single channel that is more reliable than any constituent path. The mathematics developed by Hurwitz and Radon is used to derive fundamental limits on transmission rates. The algebraic structure of the 2 × 2 space-time block code (a representation of Hamilton's biquaternions) is used to suppress interference from a second space-time user, when a second antenna is available at the mobile terminal.

Introduction

Classical coding theory is concerned with the representation of information that is to be transmitted over some noisy channel. This general framework includes the algebraic theory of error correcting codes, where codewords are strings of symbols taken from some finite field, and it includes data transmission over Gaussian channels, where codewords are vectors in Euclidean space. Fifty years of information theory and coding has led to a number of consumer products that make essential use of coding to improve reliability; for example, compact disk players, hard disk drives and wireline modems. The discovery of turbo codes by Berrou, Glavieux, and Thitmajshima [3] has led to the construction of codes that essentially achieve the Shannon capacity of the Gaussian channel.